Kavala was detained at Istanbul's Ataturk airport after flying in from the southern city of Gaziantep, the Dogan news agency said.
His lawyer Ferat Cagil told AFP that Kavala was apprehended immediately after the flight landed late Wednesday and has been remanded in custody for seven days ahead of a court hearing.
Cagil said it was unclear what his client was accused of. "The case is confidential. Whatever is said would be speculative now. We will find out when he testifies," he said.
Turkey PM says 161 dead in coup bid, almost 3,000 detained
Kavala is chairman of the Anadolu Kultur (Anatolian Culture) NGO which aims to overcome differences within Turkish society especially through culture and the arts.
It has a focus on cultural collaboration with Europe, supporting joint projects between cities and co-producing initiatives ranging from theatre to magazines.
"France, like other European countries, cooperates regularly with Mr Kavala, who is a frequent point of contact for our embassy," said French foreign ministry spokeswoman Agnes Romatet-Espagne.
"France will be watching developments in this case very closely," she added, emphasising that Turkey's international obligations were based on observing human rights and freedom of expression.
Anadolu Kultur has also sought to reach out to Armenia. Turkey has no relations with the country partly due to the dispute over the mass killings of Armenians under the Ottoman Empire, which Yerevan considers a genocide.
Richard Giragosian, director of the Yerevan-based Regional Studies Center (RSC) think tank, described Kavala as "our longtime partner in efforts to 'normalise' relations with Turkey."
"He has greatly contributed to the restoration of Armenian churches and monuments in Turkey," Giragosian told AFP.
The Dogan agency reported that Kavala had been in the city of Gaziantep, in southeastern Turkey, to discuss a project with Germany's cultural outreach organisation, the Goethe Institut.
It said he was detained as part of an investigation by Istanbul prosecutors, without giving further details.
The Hurriyet daily said Kavala was detained on the plane after it landed and police had simultaneously conducted a search of Anadolu Kultur's offices in Istanbul.
The arrest comes as rights concern intensify under the state of emergency imposed after last year's failed coup against Erdogan.
Eleven human rights activists, including the two top figures from Amnesty International's Turkey branch, will go on trial in Istanbul next week on hugely controversial terror charges.
The case has raised tensions with Sweden and Germany, whose nationals are among those detained.
In telephone talks this week, Council of Europe Secretary General Thorbjorn Jagland told Turkish Justice Minister Abdulhamit Gul the activists should be released as "pre-trial detention should be a measure of last resort."
In total 156 journalists, most detained under the state of emergency, are currently behind bars, according to the P24 activist group.
Amnesty's Turkey researcher Andrew Gardner said on Twitter that the arrest of Kavala was the "latest attack on civil society in Turkey".
Detention of civil society leader Osman Kavala in latest attack on civil society in Turkey. He should be released. https://t.co/68pMK2rbjA
— Andrew Gardner (@andrewegardner) October 19, 2017
"Osman Kavala has worked tirelessly to build reconciliation, dialogue and support the rule of law in Turkey," added Emma Sinclair-Webb of Human Rights Watch.
Osman Kavala has worked tirelessly to build reconcilation, dialogue & support the rule of law in Turkey. Release him from detention
— Emma Sinclair-Webb (@esinclairwebb) October 19, 2017
Kati Piri, the European Parliament's rapporteur on Turkey, described the arrest as "very disturbing" and said she would ask EU lawmakers "to launch [an] urgent call for his release".
Very disturbing news that Osman Kavala has been detained in Istanbul. Will propose in EP to launch urgent call for his release! https://t.co/SoY37P3fYL
— Kati Piri (@KatiPiri) October 18, 2017
Reports said Kavala was born in Paris but took over the family business when his father died. He is also the co-founder of the Iletisim publishing house.
World powers urge 'stability' in Turkey
"The arrest of our friend Osman Kavala is a turning point in the oppression of the human rights movement" under Erdogan, said Benjamin Abtan, president of the European Grassroots Antiracist Movement (EGAM).
Prominent Turkish opposition commentator Hasan Cemal wrote on the T24 website: "You [Kavala] are not alone. We are beside you! Together we will continue to fight together for the rule of law, human rights and freedom in Turkey."
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